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# Signals and Systems – Relation between Convolution and Correlation

## Convolution

The convolution is a mathematical operation for combining two signals to form a third signal. In other words, the convolution is a mathematical way which is used to express the relation between the input and output characteristics of an LTI system.

Mathematically, the convolution of two signals is given by,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( t-\tau \right )d\tau =\int_{-\infty }^{\infty }x_{2}\left ( \tau \right )x_{1}\left ( t-\tau \right )d\tau}$$

## Correlation

The correlation is defined as the measure of similarity between two signals or functions or waveforms. The correlation is of two types viz. **cross-correlation** and **autocorrelation**.

The cross correlation between two complex signals 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡) is given by,

$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x^{\ast }_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x^{*}_{2}\left ( t \right )dt}$$

If 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡) are real signals, then,

$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x_{2}\left ( t \right )dt}$$

## Relation between Convolution and Correlation

The convolution and correlation are closely related. In order to obtain the crosscorrelation
of two real signals 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡), we multiply the signal 𝑥_{1}(𝑡)
with function 𝑥_{2}(𝑡) displaced by τ units. Then, the area under the product curve
is the cross correlation between the signals 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡) at 𝑡 = 𝜏.

On the other hand, the convolution of signals 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡) at 𝑡 = 𝜏 is
obtained by folding the function 𝑥_{2}(𝑡) backward about the vertical axis at the
origin [i.e., 𝑥_{2}(−𝑡)] and then multiplied. The area under the product curve is
the convolution of the signals 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡) at 𝑡 = 𝜏.

Therefore, it follows that the correlation of signals 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡) is the same
as the convolution of signals 𝑥_{1}(𝑡) and 𝑥_{2}(−𝑡).

## Analytical Explanation

The resemblance between the convolution and the correlation can be proved analytically as follows −

The convolution of two signals 𝑥_{1}(𝑡) and 𝑥_{2}(−𝑡) is given by,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( \tau-t \right )d\tau\: \: \: \cdot \cdot \cdot \left ( 1 \right )}$$

By replacing the variable 𝜏 by another variable p in the integral of Eqn. (1), we get,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-t \right )dp\: \: \: \cdot \cdot \cdot \left ( 2 \right )}$$

Now, replacing the variable 𝑡 by 𝜏 in Eqn. (2), we have,

$$\mathrm{x_{1}\left ( \tau \right )\ast x_{2}\left ( -\tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-\tau \right )dp=R_{12}\left ( \tau \right )}$$

Therefore, the relation between correlation and convolution of two signals is given by,

$$\mathrm{R_{12}\left ( \tau \right )=x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )|_{t=\tau }}$$

Similarly,

$$\mathrm{R_{21}\left ( \tau \right )=x_{2}\left ( t \right )\ast x_{1}\left ( -t \right )|_{t=\tau }}$$

Hence, it proves that the correlation of signals 𝑥_{1}(𝑡) and 𝑥_{2}(𝑡) is equivalent to
the convolution of signals 𝑥_{1}(𝑡) and 𝑥_{2}(−𝑡).

Therefore, all the techniques used to evaluate the convolution of two signals can also be applied to find the correlation of the signals directly. Similarly, all the results obtained for the convolution are applicable to the correlation.

**Note** – If one of the signals is an even signal, let the signal 𝑥_{2}(𝑡) is an even
signal [i.e. 𝑥_{2}(𝑡) = 𝑥_{2}(−𝑡)]. Then, the cross-correlation and the convolution of
the two signals are equivalent.

- Related Articles
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